Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884969 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,44,29,32,30,35)(27,36,43,33,37,49,39,46)(28,40,45,48,41,34,42,55)(38,52,56,54,51,50,53,47);(1,3,7,4)(2,5,8,6)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(26,32)(27,37)(28,41)(35,44)(39,43)(42,45)(47,54)(50,52);(1,4,7,3)(2,5,8,6)(10,15)(11,17)(12,19)(20,24)(25,27,29,37)(26,33,32,46)(28,38,41,51)(30,43,31,39)(34,47,40,54)(35,49,44,36)(42,56,45,53)(48,52,55,50);(1,5,7,6)(2,3,8,4)(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,28,29,41)(26,34,32,40)(27,38,37,51)(30,42,31,45)(33,47,46,54)(35,48,44,55)(36,50,49,52)(39,53,43,56);(1,6,7,5)(2,3,8,4)(11,17)(14,21)(18,23)(20,24)(26,32)(28,41)(33,46)(35,44)(36,49)(38,51)(42,45)(53,56)Polycyclic presentation:
2^9|a^2=gh,b^2=fi,c^2=f,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6855568);